official manual[thumpergame.com]

Score is accumulated at every checkpoint and the very end of a level.
It depends on your performance up to that point.
These are the base score values:

• no damage +1000
  
• no misses +1000
  
• perfect turn +200
  
• crak +100
  
• kill +50

These two gain +500 on every ocurrence. Only death and restart reset this.

No misses only requires you to hit all thumps. You can crash and miss everything else.

Crashing a barrier or arc after hitting a thump and holding the button gives 1 crak.
Hitting an activated thump with shell gives 3 craks.
Pounding an activated thump with shell gives 4 craks.

Upon hitting a streak of 8 thumps the amount of collected crak will multiply.

While having a thump streak multplier, pounding onto a thump will activate the last thump within range of the shockwave.

An activated thump will appear for clearing a boss phase on first attempt.
Two will appear for clearing it with a pound on first attempt.

Every checkpoint has an activated thump.

The tendrils on the side of the track can be killed with the shockwave from pounding on a thump or the sparkling thump shot at a boss.

If you are looking to absolutely maximise your score, this guide[docs.google.com] from Cocoamix and the Thumper community goes all the way.